Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1502.00617 · Physical Review A

Quantum code for quantum error characterization

S. Omkar, R. Srikanth, Subhashish Banerjee

Published May 11, 2015• 14 citations
arXivHTMLPDF

Abstract

A quantum error-correcting code is a subspace $\mathcal{C}$ such that allowed errors acting on any state in $\mathcal{C}$ can be corrected. A quantum code for which state recovery is only required up to a logical rotation within $\mathcal{C}$ can be used for the detection of errors, but not for quantum error correction. Such a code with a stabilizer structure, which we call an ``ambiguous stabilizer code'' (ASC), can nevertheless be useful for the characterization of quantum dynamics (CQD). The use of ASCs can help lower the size of CQD probe states used, but at the cost of an increased number of operations.

DOI
10.1103/physreva.91.052309

Related papers

Ambiguous stabilizer codes and quantum noise characterization

S. Omkar, R. Srikanth, Subhashish Banerjee

A quantum error correcting is a subspace $\mathcal{C}$ such that allowed errors acting on any state in $\mathcal{C}$ can be corrected. A quantum for which state recovery is only required up to a logical rotation within $\mathcal{C}$, can be used for detection of errors, but not for quantum error correction. Such a with stabilizer structure, which we call an ambiguous stabilizer code (ASC), can nevertheless be useful for the characterization of quantum dynamics (CQD). The use of ASCs can help lower the size of CQD probe states used, but at the cost of increased number of operations.

•Feb 2, 2015•0 citations

Characterization of quantum dynamics using quantum error correction

S. Omkar, R. Srikanth, Subhashish Banerjee

Characterizing noisy quantum processes is important to quantum computation and communication (QCC), since quantum systems are generally open. To date, all methods of characterization of quantum dynamics (CQD), typically implemented by quantum process tomography, are off-line, i.e., QCC and CQD are not concurrent, as they require distinct state preparations. Here we introduce a method, ``quantum error correction based characterization of dynamics,'' in which the initial state is any element from the code space of a quantum error correcting code that can protect the state from arbitrary errors acting on the subsytem subjected to unknown dynamics. The statistics of stabilizer measurements, with possible unitary preprocessing operations, are used to characterize the noise, while the observed syndrome can be used to correct the noisy state. Our method requires at most $2({4}^{n}\ensuremath{-}1)$ configurations to characterize arbitrary noise acting on $n$ qubits.

arXiv:1405.0964•Physical Review A•Jan 15, 2015•26 citations

Theory of quantum error-correcting codes

Emanuel H. Knill, Raymond Laflamme

Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.

Physical Review A•Feb 1, 1997•1,593 citations

Quantum Subspace Verification for Error Correction Codes

Junjie Chen, Pei Zeng, Qi Zhao +2 more

Quantum error correction is pivotal in advancing toward large-scale quantum computation, and efficient verification is crucial for ensuring the high fidelity of code states. Traditional methodologies, such as state tomography, direct fidelity estimation, and state verification, either fall short in measurement efficiency, especially for large-scale systems, or are restricted to some specific states. In this work, we introduce a general framework for quantum subspace verification, enabling efficient and measurement-noise-robust fidelity estimation between a given state and the target subspace with a specified confidence level. By integrating the proposed subspace verification with direct fidelity estimation, we develop a composite protocol that significantly improves the efficiency of verifying the fidelity of general magic logical states, as demonstrated by intuitive numerical results. This improvement stems from the use of subspace verification, which leverages the knowledge of code subspaces to significantly reduce measurement costs. Additionally, we detail the construction of verification operators for typical error correction codes, including general stabilizer codes and quantum low-density parity-check codes, enabling their efficient implementation using practical local measurements. Notably, for certain codes, such as the Calderbank-Shor-Steane codes and quantum low-density parity-check stabilizer codes, we reduce the number of required measurement settings and sample complexity to a constant level using graphical methods. Our approach facilitates efficient and feasible verification of error correction codes and generic magic logical states, advancing their practical implementations on quantum platforms.

arXiv:2410.12551•PRX Quantum•Nov 10, 2025•1 citation

Energetic Costs of Subspace Quantum Error Correction

Jakub Czartowski, Felix C. Binder

Quantum error correction acts as an entropy pump, transferring noise-induced uncertainty from a protected quantum system into syndrome information stored in an auxiliary memory. Repeated operation requires this memory to be cleared which unavoidably contributes to the energetic cost of error correction. Here, we characterise this contribution for subspace quantum error-correcting codes and identify how it depends on the joint structure of the code, the noise, and the representation of the retained syndrome information. Starting from the Knill--Laflamme conditions, we construct an effective syndrome state whose von Neumann entropy sets a lower bound on the ideal work required to maintain a reusable syndrome register. Projective syndrome readout generally generates additional entropy, and we quantify the resulting gap through measurement inefficiency. We then specialise to stabiliser codes under independent local Pauli noise and analyse two classical levels of syndrome representation. At the level of abstract error labels, degeneracies among single-qubit errors reduce the leading-order entropy of processed recovery labels. At the parity-check level, lower-weight checks reduce the marginal entropy generated by individual measurement outcomes in the low-noise regime. We identify the additional burden associated with retaining and separately erasing these outcomes as a bit-level inefficiency, and illustrate both costs for the five-qubit, Steane, generalised Shor, and rotated surface codes. Our results establish a hierarchy of syndrome-memory energetic costs and identify the code, noise, and measurement structures that control the ideal thermodynamic burden of subspace quantum error correction.

quant-pharXiv:2609.03825•Sep 4, 2026•0 citations

A New Error Correction Intuitive Approach in Quantum Communications Protocols

Skander Aris, Merabtine Nadjim, Malek Benslama

Abstract: Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct errors and a formal proof that the classical bounds on the probability of errorcorrecting codes applies to error correcting quantum codes, provided that the interaction is dominated by an identity component.

American Journal of Applied Sciences•Aug 1, 2007•3 citations
A quantum error-correcting code is a subspace C\mathcal{C}C such that allowed errors acting on any state in C\mathcal{C}C can be corrected. A quantum code for which state recovery is only required up to a logical rotation within C\mathcal{C}C can be used for the detection of errors, but not for quantum error correction. Such a code with a stabilizer structure, which we call an ``ambiguous stabilizer code'' (ASC), can nevertheless be useful for the characterization of quantum dynamics (CQD). The use of ASCs can help lower the size of CQD probe states used, but at the cost of an increased number of operations.
A quantum error correcting is a subspace C\mathcal{C}C such that allowed errors acting on any state in C\mathcal{C}C can be corrected. A quantum for which state recovery is only required up to a logical rotation within C\mathcal{C}C, can be used for detection of errors, but not for quantum error correction. Such a with stabilizer structure, which we call an ambiguous stabilizer code (ASC), can nevertheless be useful for the characterization of quantum dynamics (CQD). The use of ASCs can help lower the size of CQD probe states used, but at the cost of increased number of operations.
Characterizing noisy quantum processes is important to quantum computation and communication (QCC), since quantum systems are generally open. To date, all methods of characterization of quantum dynamics (CQD), typically implemented by quantum process tomography, are off-line, i.e., QCC and CQD are not concurrent, as they require distinct state preparations. Here we introduce a method, ``quantum error correction based characterization of dynamics,'' in which the initial state is any element from the code space of a quantum error correcting code that can protect the state from arbitrary errors acting on the subsytem subjected to unknown dynamics. The statistics of stabilizer measurements, with possible unitary preprocessing operations, are used to characterize the noise, while the observed syndrome can be used to correct the noisy state. Our method requires at most $2({4}^{n}\ensuremath{-}1)$ configurations to characterize arbitrary noise acting on nnn qubits.
Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.
Quantum error correction is pivotal in advancing toward large-scale quantum computation, and efficient verification is crucial for ensuring the high fidelity of code states. Traditional methodologies, such as state tomography, direct fidelity estimation, and state verification, either fall short in measurement efficiency, especially for large-scale systems, or are restricted to some specific states. In this work, we introduce a general framework for quantum subspace verification, enabling efficient and measurement-noise-robust fidelity estimation between a given state and the target subspace with a specified confidence level. By integrating the proposed subspace verification with direct fidelity estimation, we develop a composite protocol that significantly improves the efficiency of verifying the fidelity of general magic logical states, as demonstrated by intuitive numerical results. This improvement stems from the use of subspace verification, which leverages the knowledge of code subspaces to significantly reduce measurement costs. Additionally, we detail the construction of verification operators for typical error correction codes, including general stabilizer codes and quantum low-density parity-check codes, enabling their efficient implementation using practical local measurements. Notably, for certain codes, such as the Calderbank-Shor-Steane codes and quantum low-density parity-check stabilizer codes, we reduce the number of required measurement settings and sample complexity to a constant level using graphical methods. Our approach facilitates efficient and feasible verification of error correction codes and generic magic logical states, advancing their practical implementations on quantum platforms.
Quantum error correction acts as an entropy pump, transferring noise-induced uncertainty from a protected quantum system into syndrome information stored in an auxiliary memory. Repeated operation requires this memory to be cleared which unavoidably contributes to the energetic cost of error correction. Here, we characterise this contribution for subspace quantum error-correcting codes and identify how it depends on the joint structure of the code, the noise, and the representation of the retained syndrome information. Starting from the Knill--Laflamme conditions, we construct an effective syndrome state whose von Neumann entropy sets a lower bound on the ideal work required to maintain a reusable syndrome register. Projective syndrome readout generally generates additional entropy, and we quantify the resulting gap through measurement inefficiency. We then specialise to stabiliser codes under independent local Pauli noise and analyse two classical levels of syndrome representation. At the level of abstract error labels, degeneracies among single-qubit errors reduce the leading-order entropy of processed recovery labels. At the parity-check level, lower-weight checks reduce the marginal entropy generated by individual measurement outcomes in the low-noise regime. We identify the additional burden associated with retaining and separately erasing these outcomes as a bit-level inefficiency, and illustrate both costs for the five-qubit, Steane, generalised Shor, and rotated surface codes. Our results establish a hierarchy of syndrome-memory energetic costs and identify the code, noise, and measurement structures that control the ideal thermodynamic burden of subspace quantum error correction.
Abstract: Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct errors and a formal proof that the classical bounds on the probability of errorcorrecting codes applies to error correcting quantum codes, provided that the interaction is dominated by an identity component.